FOUNDATIONS OF THEORETICAL STATISTICS. 
353 
we have, approximately, for the efficiency of v tx , 
(r + 2' ; + 1 r + 2 ... ) r— 1 r —3 r— 5 
(r 2 +r+10) r 2 7" — 2“* 
or, when r is great, 
and for the efficiency of r (X , 
28'8 . 
or, when r is great, 
(r + 2“ + f... ) r + 1~ r—5 r—7 . 
(r 2 — r +18) r r — J" r — 3 
, 53-3 
The following table gives the values of the transcendental quantities required, and 
the efficiency of the method of moments in estimating the value of v and r from samples 
drawn from Type VII. distribution. 
r. 
r + 2' j F (^) 
- 2rTl r+ 1 . 
Efficiency 
of v 
- 3-2 - -'2 
r + 1 r + 2 
‘WVMs)} 
- 2r+ 1 r + 4. 
Efficiency 
of r, x . 
5 
5-31271 
0 
6 
5-31736 
0-2572 
7 
5-32060 
0-4338 
5-9473 
0 
8 
5-32296 
0-5569 
5-9574 
0-1687 
9 
5-32472 
0-6449 
5-9649 
0-3130 
10 
5-32607 
0-7097 
5-9706 
0-4403 
11 
5-32713 
0-7586 
5-9750 
0-5207 
12 
5-32797 
0-7963 
5-9787 
0-5935 
13 
5-32866 
0-8259 
5-9810 
0-6519 
14 
5-32919 
0-8497 
5-9839 
0-6990 
15 
5-9853 
0-7376 
16 
5-9870 
0-7694 
17 
5-9883 
0-7959 
18 
5-9895 
0-8182 
It will be seen that we do not attain to 80 per cent, efficiency in estimating the form 
of the curve until r is about 17*2, which corresponds to ft 2 = 3-42. Even for sym¬ 
metrical curves higher values of (3 2 imply that the method of moments makes use of 
less than four-fifths of the information supplied by the sample. 
